x^2-8x+1=0 — step-by-step solution

1) Let's solve in

ℝ

x^{2} - 8 x + 1 = 0

Given the quadratic form

a x^{2} + b x + c

, you can calculate the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -8

and

c = 1

Δ = {(-8)}^{2} - 4 \cdot 1 \cdot 1

ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

{(-8)}^{2}

(-8) \cdot (-8)

64

Δ = 64 - 4

ⓘ Start from

64

and move back

4

on the number line:

64 - 4

60

Δ = 60

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-8) - \sqrt{60}}{2 \cdot 1}

x_{1} = \frac{8 - \sqrt{60}}{2}

ⓘ We look for the largest square dividing

60

4

2^{2}

, and take it out of the root:

\sqrt{60}

\sqrt{4 \cdot 15}

\sqrt{4} \cdot \sqrt{15}

2 \cdot \sqrt{15}

x_{1} = \frac{8 - 2 \cdot \sqrt{15}}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{8 - 2 \cdot \sqrt{15}}{2}

\frac{2 \cdot (4 - \sqrt{15})}{2}

4 - \sqrt{15}

x_{1} = 4 - \sqrt{15}

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-8) + \sqrt{60}}{2 \cdot 1}

x_{2} = \frac{8 + \sqrt{60}}{2}

ⓘ We look for the largest square dividing

60

4

2^{2}

, and take it out of the root:

\sqrt{60}

\sqrt{4 \cdot 15}

\sqrt{4} \cdot \sqrt{15}

2 \cdot \sqrt{15}

x_{2} = \frac{8 + 2 \cdot \sqrt{15}}{2}

ⓘ A fraction is simplified by dividing the numerator and the denominator by the same number:

\frac{8 + 2 \cdot \sqrt{15}}{2}

\frac{2 \cdot (4 + \sqrt{15})}{2}

4 + \sqrt{15}

x_{2} = 4 + \sqrt{15}

▶Solutions found:

x = {4 - \sqrt{15}, 4 + \sqrt{15}}

2) The numeric value of:

x = {4 - \sqrt{15}, 4 + \sqrt{15}}

equals:

x = {\approx 4 - 3.8730, \approx 4 + 3.8730}

▷

x = {\approx 0.1270, \approx 7.8730}